A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks.

Show that for any integer $n \ge 2$ the sum of the fractions $\frac{1}{ab}$, where $a$ and $b$ are relatively prime positive integers such that $a < b \le n$ and $a+b > n$, equals $\frac{1}{2}$. (Integers $a$ and $b$ are called relatively prime if the greatest common divisor of $a$ and $b$ is $1$.)

A pyramid has square base and equal sides. It is cut into two parts by a plane parallel to the base. The lower part (which has square top and square base) is such that the circumcircle of the base is smaller than the circumcircles of the lateral faces. Show that the shortest path on the surface joining the two endpoints of a spatial diagonal lies entirely on the lateral faces. [img]https://cdn.artofproblemsolving.com/attachments/c/8/170bec826d5e40308cfd7360725d2aba250bf6.png[/img]

The two wheels shown below are spun and the two resulting numbers are added. The probability that the sum is even is [asy] draw(circle((0,0),3)); draw(circle((7,0),3)); draw((0,0)--(3,0)); draw((0,-3)--(0,3)); draw((7,3)--(7,0)--(7+3*sqrt(3)/2,-3/2)); draw((7,0)--(7-3*sqrt(3)/2,-3/2)); draw((0,5)--(0,3.5)--(-0.5,4)); draw((0,3.5)--(0.5,4)); draw((7,5)--(7,3.5)--(6.5,4)); draw((7,3.5)--(7.5,4)); label("$3$",(-0.75,0),W); label("$1$",(0.75,0.75),NE); label("$2$",(0.75,-0.75),SE); label("$6$",(6,0.5),NNW); label("$5$",(7,-1),S); label("$4$",(8,0.5),NNE); [/asy] $\text{(A)}\ \dfrac{1}{6} \qquad \text{(B)}\ \dfrac{1}{4} \qquad \text{(C)}\ \dfrac{1}{3} \qquad \text{(D)}\ \dfrac{5}{12} \qquad \text{(E)}\ \dfrac{4}{9}$

A grasshopper is jumping about in a grid. From the point with coordinates $(a, b)$ it can jump to either $(a + 1, b),(a + 2, b),(a + 1, b + 1),(a, b + 2)$ or $(a, b + 1)$. In how many ways can it reach the line $x + y = 2014?$ Where the grasshopper starts in $(0, 0)$.

Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]

Let $x$, $y$, and $z$ be three real positive numbers such that $x^{2}+y^{2}+z^{2}+2xyz=1$. Prove that $2(x+y+z)\leq 3$.

The inscribed circle of $\Delta ABC$ is tangent to $AC$ and $BC$ in points $M$ and $N$ respectively. Line $MN$ intersects line $AB$ in point $P$, so that $B$ is between $A$ and $P$. Determine $\angle ABC$, if $BP=CM$.

A function $f:\{1,2, \ldots, n\} \to \{1, \ldots, m\}$ is [i]multiplication-preserving[/i] if $f(i)f(j) = f(ij)$ for all $1 \le i \le j \le ij \le n$, and [i]injective[/i] if $f(i) = f(j)$ only when $i = j$. For $n = 9, m = 88$, the number of injective, multiplication-preserving functions is $N$. Find the sum of the prime factors of $N$, including multiplicity. (For example, if $N = 12$, the answer would be $2 + 2 + 3 = 7$.)

Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.

Inside a regular triangle $ABC$, points $X$ and $Y$ are chosen such that $\angle{AXC} = 120^{\circ}$, $2\angle{XAC} + \angle{YBC} = 90^{\circ}$and $XY = YB = \frac{AC}{\sqrt{3}}$. Prove that point $Y$ lies on the incircle of triangle $ABC$.

Prove that there are infinite pairs of non-congruent triangles that have the same angles and two of their equal sides. Develop an algorithm or rule to obtain these pairs of triangles and indicate at least one pair that satisfies the asserted.

Given triangle $ ABC$. Point $ O$ is the center of the excircle touching the side $ BC$. Point $ O_1$ is the reflection of $ O$ in $ BC$. Determine angle $ A$ if $ O_1$ lies on the circumcircle of $ ABC$.

Prove the following inequality: $x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n$ where $\forall _{x_i} x_i > 0$

Suppose $N$ is any positive integer. Add the digits of $N$ to obtain a smaller integer. Repeat this process of digit-addition till you get a single digit numbem. Find the number of positive integers $N \le 1000$, such that the final single-digit number $n$ is equal to $5$. Example: $N = 563\to (5 + 6 + 3) = 14 \to(1 + 4) = 5$ will be counted as one such integer.

In Oddland there are stamps with values of $1$ cent, $3$ cents, $5$ cents, etc., each for odd number there is exactly one stamp type. Oddland Post dictates: For two different values on a letter must be the number of stamps of the lower one value must be at least as large as the number of tokens of the higher value. In Squareland, on the other hand, there are stamps with values of $1$ cent, $4$ cents, $9$ cents, etc. there is exactly one stamp type for each square number. Brands can be found in Squareland can be combined as required without further regulations. Prove for every positive integer $n$: there are the same number in the two countries possibilities to send a letter with stamps worth a total of $n$ cents. It makes no difference if you have the same stamps on arrange a letter differently. (Stephan Wagner)

Let $ n>0$ be a natural number. Determine all the polynomials of degree $ 2n$ with real coefficients in the form $ P(X)\equal{}X^{2n}\plus{}(2n\minus{}10)X^{2n\minus{}1}\plus{}a_2X^{2n\minus{}2}\plus{}...\plus{}a_{2n\minus{}2}X^2\plus{}(2n\minus{}10)X\plus{}1$, if it is known that all the roots of them are positive reals. [i]Proposer[/i]: [b]Baltag Valeriu[/b]

Let $n\geq 2$ be integer. On a plane there are $n+2$ points $O,\ P_{0},\ P_{1},\ \cdots P_{n}$ which satisfy the following conditions as follows. [1] $\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).$ [2] $\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$

Consider the polynomial \[P(x)=x^3+3x^2+6x+10.\] Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$. [i]Proposed by Nathan Xiong[/i]

Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.

Solve the system of simultaneous equations \[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\] in real numbers.

Find the real solution(s) to the equation $ (x \plus{} y)^2 \equal{} (x \plus{} 1)(y \minus{} 1)$.

Given are $n$ pairwise intersecting convex $k$-gons on the plane. Any of them can be transferred to any other by a homothety with a positive coefficient. Prove that there is a point in a plane belonging to at least $1 +\frac{n-1}{2k}$ of these $k$-gons.

Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overline{DE}$, and $\gamma_3$ is tangent to side $\overline{EF}$ and side $\overline{FA}$. Then the area of $\gamma_1$ can be expressed as $\frac{m\pi}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$. [i]Proposed by Sean Li[/i]

Suppose $x_1,x_2,x_3,\dotsc$ is a strictly decreasing sequence of positive real numbers such that the series $x_1+x_2+x_3+\cdots$ diverges. Is it necessary true that the series $\sum_{n=2}^{\infty}{\min \left\{ x_n,\frac{1}{n\log (n)}\right\} }$ diverges?